Continuous-state branching processes with collisions : first passage times and duality

Foucart, Clément (author), Vidmar, Matija (author)

We introduce a class of one-dimensional positive Markov processes generalizing continuous-state branching processes (CBs), by taking into account a phenomenon of random collisions. Besides branching, characterized by a general mechanism $\Psi$, at a constant rate in time two particles are sampled uniformly in the population, collide and leave a mass of particles governed by a (sub)critical mechanism $\Sigma$. Such CB processes with collisions (CBCs) are shown to be the only Feller processes without negative jumps satisfying a Laplace duality relationship with one-dimensional diffusions on the half-line. This generalizes the duality observed for logistic CBs by Foucart (2019). Via time-change, CBCs are also related to an auxiliary class of Markov processes, called CB processes with spectrally positive migration (CBMs), recently introduced by Vidmar (2022). We find necessary and sufficient conditions for the boundaries $0$ or $\infty$ to be attracting and for a limiting distribution to exist. The Laplace transform of the latter is provided. Under the assumption that the CBC process does not explode, the Laplace transforms of the first passage times below arbitrary levels are represented with the help of the solution of a second-order differential equation, whose coefficients express in terms of the Lévy–Khintchine functions $\Sigma$ and $\Psi$. Sufficient conditions for non-explosion are given.

• Language: English
• Keywords: continuous-state branching process, branching process with interactions, first passage time, Laplace duality, Lamperti time-change, one-dimensional diffusion
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: 35 str.
• Numbering: Vol. 167, art. 104230
• PID: 20.500.12556/RUL-153109
• UDC: 519.2
• ISSN on article: 0304-4149
• DOI: 10.1016/j.spa.2023.104230
• COBISS.SI-ID: 170650115
• Publication date in RUL: 18.12.2023

Record is a part of a journal

• Title: Stochastic processes and their applications
• Shortened title: Stoch. process. their appl.
• Publisher: Elsevier
• ISSN: 0304-4149
• COBISS.SI-ID: 26454016

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Projects

• Funder: ARRS - Slovenian Research Agency
• Project number: N1-0174
• Name: Časi prvih prehodov enostranskih procesov Markova

• Funder: ARRS - Slovenian Research Agency
• Project number: P1-0402
• Name: Matematična fizika